# Abstract Confusions

Complexity is not a cause of confusion. It is a result of it.

## Urinal Protocol and Hilbert’s Hotel

There are two things I read this week and thought weird. One is about ‘Urinal protocol’.

Urinal protocol vulnerability

For those who don’t know what it is, read here. The point is, in a public rest room, every male should take due effort to use the buffer urinal. There are un-written rules, protocols people follow in using urinals. The common one being, using the urinal which is at the maximum distance to the one occupied already. Then, the point of using, allocating these resources effectively comes in as a concern.

• Is it possible to know / determine, the optimum number of urinals? So that, you will built the optimum numbers so that they are used efficiently.
• Are there any bad numbers? good numbers? any study on it? Want to know for what number there will be maximum occupancy, and less occupancy.
• Any civil contractor who is building a public building / mall / college wants to know what is the appropriate number of urinals to built? I guess there is no study available. Now we have one.

It happens to be that, there is a detail study available on urinal protocol vulnerability. I have already mentioned about XKCD here.

Urinal protocol vulnerability - Courtsey - blag.xkcd.com

He goes further and generalises, for how many urinals, the protocols work good and for what numbers bad.  Here are the results.

The good numbers are given by equation $n=2^k + 1, k \in I$ results in $n=3,5,9,17...$. When you have urinals in these numbers, and if people follow the urinal protocol, you will atleast get 50% occupancy.

And bad numbers are given by, $n=\frac{3}{2}\times 2^k + 1, k \in I$ gives $n=4,7,13...$. These number of urinals have only 33% occupancy.

Hilbert’s Hotel

Hilbert’s hotel is a nice paradox on set theory. Read here for more on this. And a nice explanation here.  It is about an important concept in set theory. It is told often that infinity is not  different from cardinality. Although natural numbers are infinite, the cardinality of natural numbers is lesser to cardinality of real numbers. This is explained using $\aleph_0, \aleph_1 ... \aleph_\omega$ notations.

The story goes like this. Assume of a hotel. A traveler enters the hotel, goes to the receptionist and asks for a room. The receptionist tells, they have aleph-null rooms and all of them are occupied. It doesn’t mean, he can’t get room to stay. The receptionist says, he can move the person staying in the first room to second room, second room person to third room and so on. Then he can occupy the first room. This is fine. This is true with aleph-null hotel.

The paradox goes on, if a bus full of travelers come asking for a room? Say, aleph-null number of traveler’s? Then Hilbert says, you can occupy them by shifting, the 1 room person can be asked to move to 2, move 2nd person to 4, move 3rd person to 6, move 4th person to 8, move 5th person to 10… and so on. Once this is done, you will have all odd numbered rooms empty. And the aleph-null bus travelers can occupy them.

The idea is really difficult to get through. But that’s how abstract concepts are placed in math.

And strangely, I have read two entirely different concepts, relating to occupancy. One is used in every day and another one not so easy to even think off. That’s the beauty of math.

### 7 responses to “Urinal Protocol and Hilbert’s Hotel”

1. Akhil Mathew October 14, 2009 at 12:24 PM

The Hilbert hotel argument in question has applications in, e.g., the Eilenberg swindle, which I posted about at

• SREE GURUPARAN October 17, 2009 at 1:03 AM

I tried reading the post. But I should admit that I could not proceed much. Perhaps, I need to read my algebra book again. BTW, the blog is really cool. I need to read more of them. Thnx.

2. Savannah October 17, 2009 at 3:42 AM

Awesome blog!

I thought about starting my own blog too but I’m just too lazy so, I guess Ill just have to keep checking yours out.
LOL,

3. jerrymaguire October 20, 2009 at 8:18 PM

I was at Smokey Bones the other day and saw the greatest invention I have ever seen! It was a football field goal post with a football attached to urinal screen in the urinal and it was genius! Guys aim at it while relieving themselves. You have to see it yourself, the website of the company that makes them is at http://www.direct-aim.com and every sports bar should have these!

4. john @ john October 30, 2009 at 7:30 PM

thanks for the post